S1-Equivariant Function Spaces and Characteristic Classes
Author(s): Benjamin M. Mann, Edward Y. Miller, Haynes R. Miller
Source: Transactions of the American Mathematical Society, Vol. 295, No. 1 (May, 1986), pp.
233-256
Published by: American Mathematical Society
Stable URL: http://www.jstor.org/stable/2000155 .
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J-homomorphisms
mology
composing
Schultz ofspace
equivariant
the
we J-homomorphisms
obtain
SG(S1)
Jz2:
''S1-transfers
t:
U
Q(CP+°°
SO
QS°
(E/H)¢H
SG(Z2)
in/\CP+°°
homology
SO
S1)Q(RP+°°)
and
Q((E/K)¢K
of stable
/\
S1
answering
isl
Q((BZpn)+)
and
Sl-equivariant
UU)SG(S1).
aQ(CP+°°
question
n > 0 /\
and
self-maps
ofS1).
ofJ. the
P.
By
TRANSACTIONS OF THE
AMERICAN MATHEMATICAL SOCIETY
Volume 295, Number 1, May 1986
S1-EQUIVARIANTFUNCTION SPACES
AND CHARACTERISTICCLASSES
BENJAMIN M. MANN EDWARDY. MILLERAND HAYNESR. MILLER
ABSTRACT. We determine the structure of the homology of the Beckerof spheres (with standard free S1-action) as a Hopf algebra over the DyerLashof algebra. We use this to compute the homology of BSG(S1). Along
the way we give a fresh account of the partially framed transferconstruction
and the Becker-Schultzhomotopy equivalence. We compute the effect in ho-
May.
Introduction. Let H be a compactLie groupadmittinga finite-dimensional
W suchthat H acts freelyon the unit spheresW. H
orthogonalrepresentation
of S1 in S3) oroneof a knownlist [13]of finite
mustthusbe S1,S3) the normalizer
including(as subgroupsof S3) the cyclicand
groupswith periodiccohomology,
generalized
quaternion
groups.Let EndH(sW)denotethe spaceof H-equivariant
continuous
self-mapsof sW. Byjoiningwiththe identitymapwe obtaininclusions
EndH(s(nW))C EndH(s((n+l)W)), andwewriteG(H) forthe directlimit. The
by J. C. BeckerandR. E. Schultz[2],and
homotopytypeof G(H)wasdetermined
of G(H)
of W. If we writeSG(H) forthe component
turnsout to be independent
= G(H)if H is connected),then in [3],
containingthe identitymap (so SG(H)
productin
BeckerandSchultz(seealso[9]in caseHis finite)enrichthecomposition
spaceBSG(H)
classifies
spacestructure.Theclassifying
SG(H)
to an infinite-loop
H-actionmodelledon s(nW),
orientedspehricalfibrationswitha fiber-preserving
withfibersW.
joinswiththe trivialH-fibration
stabilizedby formingfiberwise
In this paperwe determinethe modphomologyof SG(S1)andof BSG(S1) as
algebra.Alongthe way,we computethe effect
Hopfalgebrasoverthe Dyer-Lashof
and of the equivariant
mapsSG(S1) ) SG(Zpn)
in homologyof the "forgetful"
maps. §1
The startingpointforouranalysisis the studyof certain"transfer"
of thesemaps.
andgeneralproperties
is devotedto an accountof the constructioll
In §2we studycertaintransferst associatedto an inclusionK c H of compactLie
groups.If E is a smoothprincipalH-space,then
obtainedby mixingE 1 E/H with the adjoint
whereW;H
is the vector-bundle
denotesformationof the
representation
of H on its Lie algebra,the superscript
infiniteloopspaceof X.
ThomspaceandQX is the enveloping
Received by the editors August 12 1980 and in revised form June 15 1985.
1980 Mathematics Subject Classification. Primary55R40; Secondary55R91 55Q50 55R12.
({)1986 American Mathematical Society
0002-9947/86 $1.00 + $.25 per page
233
tl*
iZ2
is1:
t ty:
H*(Q(CP+
SO
U5EndH(E)
)) G(H)
SG(S1)
SG(Z2)
A Q((E/H)¢H
Q(BH¢)
S1))
Q(CP+
QRP+
H*(QBZ+)
AX)S1)
B. M. MANN, E. Y. MILLER AND H. R. MILLER
234
Thenin §3we describea map
generalizing
a variantof a construction
of Beckerand Schultz[2]. Weshowthat
if K c H)then the inclusionEndH(E)c EndK(E)corresponds
undertyto the
transfert. If H is a "periodic"
Liegroupas above,thenwe obtainfromtya map
whichwasshownby BeckerandSchultzto be a homotopyequivalence.
In particular,SG(S1)= G(S1)is homotopyequivalent
to Q(CP+AS1).
§4is devotedto the evaluationin modphomologyof the transfer
tn: CP+ AS1 ) Q(BZ+n)) n > O,
associatedto the inclusionZpnc S1.Thisis oneof ourprincipaltechnicalresults.
Wegivethestatementhereforn = 1. Letar E H2r+1tCP+ASl) ander E hr(BZp)
be the standardgenerators(see §4),andlet X be the canonicalantiautomorphism
on H*(QX)
THEOREMA .
Forp = 2,
tl*ar = E
e2>+l * xe2(r_r) + E
Qt+let * (xer_t)*2.
Forp > 2,
tl*ar = E
e2>+l * xe2(r_>).
Sincetransferscomposewell,andthe transferassociatedto 1 c Zpessentially
definesthe Dyer-Lashof
operations,
it is easyto deducefromTheoremA the effect
of to in homology;
the formulaearegivenin TheormeC, andin Theorems4.4 and
4.5. An easyfiltrationargument(carriedout in §7)resultsin the
C OROLLARY
. Forp =
2,
t0*:
H*(Q(CP+
A S1))
) H*(QS°)
is injective.Forp odd,
is injective,whileto*is not.
§5is dedicatedto a studyof the equivariant
J-homomrphisms
Let
Ac: CPn-1ASl U(n)
senda pair(1,z), whereI c Cn is a complexline and z E C has lZl = 1, to the
unitarytransformation
whichis the identityon 11 andmultipliesby z on 1. Then
we havethe followingtheorem.
plex J-homomorphism
Jc: U QS°in termsof the loop-structure in H*(QS°).
S1-EQUIVARIANT FUNCTION SPACES
THEOREMB.
235
Thecomposite
CP+°°AS1 >> U is1 Q(CP+°°
AS1)
is homotopic
to thestandardinclusion.
Thereis a realanalogue;
seeTheorem5.1. Theorems
A andB haveas a corollary
the determination
of the imageof the generators
of H*(U)underthe classicalcomForp odd,this resolvesa problemleft openby J. P. Mayin [5, pp. 121-123].
THEOREMC. Theimageof arE H2r+l(u;
Zp) underJc* is
Jc*ar =
Q
E
+
E
[1]* XQ ( 8)[1]* [1]
Qt+lQt[l]
*
(XQr
t[l])
*
[1]
for p = 2, and
Jc*ar
= (-l)k
Es
dQ8[1]* XQk-8[l]
=O
if r = (p- 1)k- 1
otherwise
for p odd.
Theinfinite-loop
structureof SG(H) is verydifferentfromthe naturalinfiniteloop structureon Q(BH¢H). In [11], R. E. Schultzdescribedthe composition
productin termsof the transferassociatedto the
diagonal
embedding
/\:
H
H x H. ForH finite,this theoremis recoveredandextendedin [8];and by the
Corollary
to TheoremA, this casesufficesforpresentpurposes.In §6we combine
this with TheoremA to provethe followingresult,whichcompletelydetermines
the composition
producto in H*(SG(S1)).
THEOREMD. Forp = 2,
aqo ar = aq* ar + (q
+ (q-2t,
- s, r - s)Q2(q+r-8)+la
r-2t)Qq+r+lqq+r-2ta
Forp odd,
aqo ar = aq* ar + E c(q,r,t):Qtaq+r+1_(p_l)t,
t
where
(q) ) )
#(
)
(
r
) (
t- k
)
Wethenappealto resultsof [8],relatingG(H) forH finiteto the multiplicative
structurein a certainEOO-ring
spaceA(H), the "Burnside
space"of H, introduced
(as a space)by G. Segal[12]. By the Corollaryto TheoremA, the relationship
betweenthe Dyer-Lashof
operationsin SG(S1) andthe *-productandloopDyerLashofoperationsin Q(CP+°°
AS1) areimpliedby analogousrelationships
forZp.
Theseformulae
showhowtheDyer-Lashof
actiononar (given,inviewofTheoremB
B+
A Sk- B- Elo(i)
B. M. MANN, E. Y. MILLER AND H.
R. MILLER
236
andthe factthat iS1 is an infinite-loop
map,by Kochman's
formula[7])determine
the actionon all of H*(SG(S1)).
We turn next to a "global"analysisof
H*(SG(S1))by meansof a "weight
valuation".In §7,we prove
THEOREME.
P= 2
H*(SG(S1);Zp)is a primitivelygenerated
Hopfalgebra.For
H*(SG(S1))-H*(U) X P[ar2: r > 0]
@ H*(Q(CP+ AS1))//P[ar:
r > O],
andfor p odd,
H*(SG(S1))- H*(Q(CP+ AS1))
as Hopfalgebras.
Finally,we follow[10, 5 and8] in usingthe
classifying-space
spectral
andDyer-Lashof
operationsto prove(usingthe usualDyer-Lashof sequence
notation[5, p.
16])
As Hopfalgebras,
for p = 2
H*(BSG(S1))- H*(BU) X E[aQ2r+lar:
r > O]
@ p[Q2r+2ar:
r > O]
XP[aQIar:
I admissible,I(I) > 1, e(I) > 2r+2, r > O],
andfor p odd,
H*(BSG(S1))- H*(BU)X S[a/3£Qr+lar:
E = O or 1, r > 0;
THEOREMF.
I admissible,I(I) > 1, e(I) + b(I) > 2r+ 2, r >
0;.
Here
S denotesthefreecommutative
algebra.
Wewishto thankJ. P. Mayforan instructive
correspondence
aboutsigns.This
work
wasdonein 1977-1979,whenthe authorswere
at
Harvard
University(Mann
andH. Miller)and M.I.T.(E. Miller). During
this time all threeauthorswere
partially
supportedby the NSF.
@ S[aQIar:
1. Generalitie#on the tran#fer. In this
sectionwe recallthe definition
thetransfer,generalized
slightlyto allowtwistingby a vectorbundle. We of
then
catalogue
variousproperties
whichwillbe usedlater.All manifolds
will
be
smooth,
and
we writer(M) forthe tangentbundleof M.
Letlr: E ) B be a smoothmapbetweenclosed
j:E > -B of E into the trivialvectorbundle manifolds.Chooseanembedding
overB of fiber-dimension
k, such
that
(1.1)
X ",
/
pr
B
commutes.
(Forexample,embedE intoRk by i, andlet j have
components
(lr,i).)
Write
zJ(;)for the normalbundleof j. The
Pontrjagin-Thom
construction
then
yields
a mapof Thomspaces:
(1.2)
(1.5)
t1r
= t1r,+:
EX¢@>(i)Ed@T(X)@v(i)
Bf Ee
S1-EQUIVARIANT FUNCTION SPACES
237
Givenany vectorbundle¢ overB, we may composej with the axis embedding
) ¢ @ kB; the Pontrjagin-Thom
construction
then givesa "twisted"
formof
_B
(1.2)
(1.3)
Bf ASk- B¢@- , Es ¢@v(i).
Givenalsoa bundle4 overE, a relativeframingof lr,(, ¢ is a bundleisomorphism
(1.4)
0: X f @ V(j) -(
@ kE
(or,rather,an equivalence
classof pairs(j, p)). Wethenobtaina stablemap
calledthe transferassociatedto 7r,(, ¢ (with(j, ) understood).
If lr is a submersion,
we havea naturalexactsequenceof vectorbundles
(1.6)
0 , r(X) , r(E) X*T(B)
,O
in whichr(lr)is the bundleof tangentalongthefiberof lr. Foranyembedding
j as
in (1.1),thereis a naturalexactsequence
°
) 7(T)
)-E
) >(i)
)°
Since-E
k hasa naturalmetric,we havea naturalsplitting:
(1.7)
7(7T)
@ I>(j)--E
A relativetrivialization
of lr,(, ¢ is a shortexactsequence
(1.8)
0
(T*¢r(T)
) O.
A choiceof metricin lr*¢splits(1.8)andgivesan isomorphism
(1.9)
GT f-4
@ 7(7T).
Combining
this with (1.7),we havedefineda relativeframing
(1.10)
4: lr*¢@IJ(j)-(
@ T(7T) @ I/(j)-(
@-E
The homotopyclassof the associatedtransfertX,¢,:Bf ) Ed is independent
of
choiceof metricin lr*¢,becausethe homotopyclassof the homeomorphism
is. All the relativeframingswe dealwithherearisein this way.
Wenownoteseveralfeaturesof the transfer.
Note1.11. Fundamental
classes.In caseB is a singlepointand¢ = O,the map
(1.3)givesa stablehomotopyclass
[E]ESn(E
(
)))
wherezo(E)
is thezerodimensional
virtualnormalbundleofE andn is thedimension
of E. Thisis the stablehomotopy
fundamental
classof E, withtwistedcoefficients.
A framingof E givesriseto a class[E]E Sn(E);andpinchingE to a pointproduces
a classin the stablehomotopyof spheres,Sn(*) namely,the usualPontrjaginThomclassof the framedmanifoldE.
fiber
Note
F,1.16.
1.15.
andthat
Pull-backs.
Compositions.
we
t1,>
are Bf
given
Let
Let
Bf
f:alrl:
A
B'
trivialization
B+
BE1
be
1AJ>+
E,
a smooth
41,Bf( zJ(j)
A
be
map
S°atransverse
relatively
(kBf
- n)E,
to
nframed
=lr,
dimF.
and
map
Then
let
to
238
B. M. MANN, E. Y. MILLER AND H. R. MILLER
Note 1.12. Fundamental
class of a framedmap. In case ( = 0 and dim¢-=
dimE - dimB = n, the transfer(1.5)composedwiththe pinchmapE ) * gives
a stablecohomotopy
class
e(7r) E S-n(B¢-n)
This is the stablefundamentalclassof lr, with twistedcoefficients.A framing
¢ @ k- n @ k givesriseto a classe E S-n(B), the Pontrjagin-Thom
classof the
framedmapir.
Note1.13. Composition
withtheprojection.Supposethat lr is a fibrationwith
F has a well-defined
framing,andso defines[F]: Sk
B¢fE3-
) sk-n.
t
EX f@(k
[ ]
Bf ASk-n
11
Thenthe diagram
n)
1s
Bf ASk
commutes.
Note1.14. Products.Let lr':E' ) B', (',¢' be anotherrelativelyframedmap.
Thenlr x lr',( x (', ¢ x ¢' has a naturalrelativeframing,andthe diagram
(B x B)¢x¢
11
Bf AB'¢'
(E x
t,
E')(X('
11
Ed AE'(
twAtw,
commutes.
E. Thenlro lrl: E1 ) B, 41,¢ hasa canonicalrelativeframing,andthe associated
transferis givenby the composite:
tXoX1
= tX1o tX
7r':E' ) B', (', ¢' be the pull-back
of lr,(, ¢ alongf. Thenlr',(', ¢' hasa canonical
relativeframing,andthe diagram
B'¢
t/
f
l
E'(
Bf
lt7r
f
Ef
commutes.
Note1.17.Reframings.
Thetransfercanbe interesting
evenin caseE = B, lr=
id, and( = ¢. Onethenhasthe canonical
framing: ¢ ) ¢ by the identity.Then
a smoothmapA:B ) O(k)inducesa newrelativeframing
1 @A: ¢@_
¢@_,
where)v(b,v) = (b,)v(b)v),withan associatedtransfert1 : Bf ) B¢. If J: O(k)+
) S° is the stablemapadjointto the inclusionO(k)c O c QS°, thentl X is the
composite
E'
(2.2)
(2.3)
Now
) E.
letThen
O
0 Kc*(E,
0H
t:
fKbe
fHH)
°fK
a fH
(E/H)¢H
closed
*f ) X(T(E)/H)
T(E)/H
(E)/K)
subgroup,
describes
fH
)(T)
soE+.
r(E/H)
athat
functor
x*7r:°,(E/K)
)TE/K
into
O.(E/H)
E/H
the
g isstable
a fiber
oO homotopy
bundle
S1-EQUIVARIANT FUNCTION SPACES
239
of moregeneralmapsmay
UsingNote 1.15,the transferassociatedto reframings
alsobe expressedin termsof J.
2. Transfersa##ociatedto Lie groups. Let H be a compactLiegroupand
E a compactsmoothprincipalrightH-space.H acts alsofromthe left on its Lie
algebrafi by the adjointaction;andE/H thussupportsa canonicalvectorbundle
E XH e ,
calledthe adjointbundle.In this sectionwe showhowthe transferendowsthe
andapplythe notesof §1
formationof the ThomspaceEf withnewfunctoriality,
whichwillbe usedin §4.
to verifysomeproperties
To begin,onemayeasilycheckthe followinglemma.
¢
= fH =
2.1. Thereis a naturalshortexactsequenceof vectorbundlesover
LEMMA
E/H
diagramof vectorbundlesover
with fiberH/K. Wehavea naturalcommutative
E/K, withexactrowsandcolumns:
o
T(T)
1
1
1
11
1
o
Theserpentlemmathenprovidesa naturalshortexactsequence
(1.8)of 7r,fK,
i.e., a relativetrivialization
fH.
map
Thereresultsa stable"transfer"
t: (E/H)¢H ) (E/K)¢K.
If, forexample,K is the trivialsubgroup,then
extendedfunc(E/H)¢Hwiththe following
Thesemapsprovidetheconstruction
tionality.Considerthe categorywhoseobjectsarepairs(E, H), whereH is a Lie
groupand E a compactsmoothprincipalH-space,in whicha morphismfrom
smoothmap
(E, H') to (E, H) is a closedinclusionH' O H andan H-equivariant
category.
we maycontructa stablemap
BH by manifolds,
By approximating
(2.4)
t: (BH)¢H
> (BK)¢K
compact
Let
the
inclusion
j: Esmooth
) (Eon
xprincipal
the
E)/H
(E/H)
first
(E/K)¢K
(E/K
be(E/K)¢K@T(X)
(E/K)¢K
H-space
fH
the
factor.
E/K
xAmap
E/K
Sn
H/K)¢KX_
A
with
xeNote
(E/H)
(H/K)+
)AH/K
base
(e,
that
SnfH
E/H
*)H
point
(E/H)¢H
j*
(E/H)¢H
(E/K)¢K
E/K
(E/K)¢K
A(¢H)
and
(E/H)
* and
let
_ and
projsetion
fH
i: i*E/H
(¢HE/H
X1r:
fH)
x E/H
EfHE/H.
@_
be
240
B. M. MANN, E. Y. MILLER AND H. R. MILLER
froma closedinclusionK C H whichis well definedup to weakhomotopy.
This givesthe suspensionspectrumof (BH)¢Ha contravariant
functoriality
on
the cateogryof Liegroupsandclosedinclusions.
Wenowprovethreelemmaswhichwillbe usefullater.
Let K c H be a closednormalsubgroup,and E a smoothprincipalH-space.
Then
(2.5)
l pr1
l
T
is a pull-backdiagramif oeis the H/K-actionmap oe(xK,h) = xhK. This action lifts to makefK an H/K-equivariant
vectorbundleoverE/K; so we havea
canonicalbundleisomorphism
°t fK-fK
XO
overE/K x H/K. Thusthe pull-backproperty(Note1.16)of the transferasserts
thatthe diagram
(2.6)
1 tprl
1 ts
is homotopy-commutative.
Nowlr is a principalH/K-bundle,so a choiceof orientationforH/K determines
a naturaltrivialization
forr(lr). NowusingNotes1.14
and 1.11,the diagram(2.6)leadsto the followinglemma.
LEMMA
2. 7. LetK c H be a closednormalsubgroup
of codimension
n, and
let E bea compactsmoothprincipalH-space.Then,withthe abovenotations,the
followingdiagramis homotopy
commutative.
1 1A[H/ff]
1 ter
Here[H/K] is theclassof H/K withthechosenorientationandthe corresponding
right-invariant
framing. g
this is a degenerate
formof a "doublecosetformula",
andits propergeneralizationshouldbe of greatinterest.
LEMMA 2.8. Let H be a compactLie groupof dimensionn, and let E be a
Thediagonalinclusion/\: H ) H x H iduces/\: (E x E)/H
Undertheseidentifications
1 terAl
E+ ASn
commutes
up to homotopy.
l tz
2
((E x E)/H)¢H
) E/H x E/H.
andthe inversion
BH¢ABHF
x X-1inE/H
H
BH
induces
lAt E/HXE/H
BHXBH
a self-map
BH¢AS° X of
BH.H
BHFacts
trivially on
241
S1-EQUIVARIANT FUNCTION SPACES
PROOF. Thediagram
E
A
(EXE)/H
1t
1s
so Note 1.16givesthe result. g
is a pull-back,
,u:HXH )H.
Nowlet H be anAbeliancompactLiegroupwithmultiplication
the classifyingspaceBH inheritsa product,i,
Since,uis a group-homomorphism,
,u is
its Lie algebra,so the bundle¢ has a canonicaltrivialization.Furthermore,
coveredby a bundlemap,i: _ x ¢ ) ¢, includingan actionmap
H BH+ABHF
,BH¢.
Lett: BHF , S° be thetransferassociatedto theinclusionof thetrivialsubgroup.
Thefollowinglemma(dueto R. Schultz[11]in the caseH is finite)computesthe
in termsof t.
transfert<, associatedto the diagonalsubgroupHCHXH,
is weakhomotopy2.9. Withtheabovenotation,thefollowingdiagram
LEMMA
commutative:
(B(HXH)¢HXH
BH¢ABHF
1 hA1
BH¢ABH+ABHF
l tz
1 lAX+Al
BH¢ABH+ABHF
1 1Ay
Applythe compositionproperty(Note 1.15) of the transferto the
diagram
commutative
PROOF.
il
X
1 a
BHXBH
wherea(x, y) = (x,x(x)y) andi1(x) = (x, 1). Of coursethe transferassociatedto
a is just the inverseof the mapa inducedon Thomspaces;so
the diffeomorphism
ta =ti1 oa.
Nowthe left leg of Lemma2.9 is a, andthe bottomcomposite,by Note 1.14,is
til.
O
construction
3. The Becker-Schultzmap. Wenowdescribethe "graph"
ty:EndH(E) ) Q((E/H)¢))
of Beckerand Schultz[2]. HereH is a Lie
whichis a variantof a construction
group,E a compactsmoothprincipalH-space,EndH(E)denotesthe spaceof Hinfiniteloopspace
conditionsself-mapsof E, andQX is the enveloping
equivariant
of the spaceX.
f'(e)
(3.2)
aEndH(E),
sociated
map
The=B
reduced
(e,f(e)).
>
Ss
to
in
EndH(E)+
BH
the
such
(E!H)
classifying
diagonal
canonical
This
a way
(E/ff)>K
H
isi\:
that
Aframing
equivariant,
tty:
Ef':
Bv
B
B,
(E2/H)Pr*>
the
E2/H
EndH(E)+
Bv
to
f(E2/H)Prlt9.
(2.2)
reuslting
(E2/ff)pr;>K
has
(E2/H)prllo
obtain
and
normal
of 1r:
QBH¢.
awe
equivariant
map
get
bundle
E/K
a (E/H)¢H
aE/H.
map
(E/K)¢K
isomorphic
f':
map
Bf:E2/H
A
Ato
M
Ss
Sxof
7(E)/H,
Eorbit
E is
242
B. M. MANN, E. Y. MILLER AND H. R. MILLER
Choose an embeddingof B = E/H in a Euclideanspace R8, and let Ivdenote
the normalbundle. We have the usual Pontrjagin-Thomcollapse
c:Ss yB>.
(3.1)
Let f: E -->
E be an equivariantmap, and form the "graph"f': E
>
E2 by
spaces. If pr1: E2/H > B is induced from projectionto the Srst factor, then
prl o f' = 1. Thus f' is coveredby a bundlemap v >prlao,and we get a map
_
on Thom spaces. This constructiondependsonly on continuityof f, and depends
continuouslyon f, so we get a map
which, by Lemma 2.1, is isomorphicto r(B) (E3
¢. Since A prlao= >, we get a
transfer
(3 3)
ti,:
(E2/H)PrlM
-->
B>@T(B)@3¢Bf ASs
The compositeof EndH(E)+A (3.1), (3.2), and (3.3), gives a pointedmap
ty: EndH(E)+ A Ss > Bf A Ss.
The map tyis obtainedby composingthe adjointof tYswith the inclusion
Q8S8Bf
QB¢,and is, up to homotopy,independentof choices. We may also composewith
The followingnaturalityresult is due essentiallyto Beckerand Schultz[2, 5.16].
THEOREM3.4. LetK bea closedsubgroup
of theLie groupH, andlet E bea
compactsmoothprincipalH-space.Then
EndH(E)
l inc
a
Q((E/H)¢H )
1 t
EndK(E)
5
Q((E/K)¢K)
commutesup to homotopyif t is the infinite-loop
extensionof the transfert asPROOF. BY Notes 1.12,1.15,and 1.16, we have a homotopy-commutative
diagram (in whichthe subscriptsindicatethe groupsinvolved)
CK \
l
t
l
t
l
t
fromwhichthe theoremfollows. O
We will usually study ty by mappingsome compact smooth manifoldM into
smooth.
givenby f'(m,
Moreover
e) =we
(f will
(m, e),
arrange
e) is transverse
that the reduced
to A: B E2/H.
graph
Then
f': the
Mxpull-back
B E2/H
S1-EQUIVARIANT FUNCTION SPACES
243
property(NoteI.16),appliedto the pull-backdiagram
r
A
B
41
(3 5)
p
1 A
{ Mx B
\
E2/H
t
1 prl
M
showsthat
M+
2 Qrs*¢
1
f Qs
EndH(E)+ <
QB
commutesup to homotopy.
In [2],BeckerandSchultzconsiderprincipalH-spacesarisingas the unitsphere
sV of an orthogonal
representation
V of H. Theyconstructa map
A: EndH(sV)+
> Q(sV/H).
The description[2, 5.6] of A showsthat it is identicalto tyexceptthat i\ is
replacedby the antidiagonal
A-(u) = (u,-u); the identification
(Theorem3.4) is
essentiallythe transfert\-. Let A(u) = -u denotethe antipodalmap. Wehavea
commutative
diagram,foranyf E EndH(sV),
(E2/H)Prlv
f jt
(3.6)
\
Bl'
tzs-
1 lxA
Af-
\-,
Bf ASs
/7
tiv
(E2 /H)Prlv
so we findthat
EndH (sV)+
(3 7)
1 (Ao)
QBf
/X r
EndH (sV)+
commutesup to homotopy.
WhenH admitsa finitedimensional
orthogonal
representation
V suchthat the
unit spheresV is a principalH-space,the mapseyare compatibleunderjoining
withthe antipodalmap,andgiven,in the notationof the introduction,
a map
(3.8)
q: G(H)
> Q(BH¢)*
THEOREM3.9 (BECKER-SCHULTZ[2]).
Wereferthe readerto [2] forthe proof.
tyis a homotopy-equivalence.
(4.1)
(4n2)
in dimension
r.(QBH¢)2
Embed
to: Zpn
CP+°°
/\2 into
CP+°°
(QBH¢)4
S1 so that
1x#xl
theAresulting
(QBH¢)3
AS1
S map
QBHf
1 7r:QS°
Ln
S°.CP°°pulls
.
244
B. M. MANN, E. Y. MILLER AND H. R. MILLER
THEOREM3.10. LetH = id. If dimV is even,then(f) = 1- [f] whereas
-if
dimV is odd,then(f ) = 1 + [f].
PROOF. [2, 6.13] showsA(f) = 1- [f]. (Thereis amistake in the original
proofof this fact in [2] but Becker(privatecommunication)
has givena correct
proof.) The correctedproofshowsthat A(f)= 1+ [f] whendimV is odd. The
evendimensonal
casefollowsdirectlyfrom(3.7) and [2, 6.13].
Finally,werecallR. E. Schultz'sexpression
forthe composition
pairingo: G(H)
xG(H) ) G(H). Let i\: H ) H x H be the diagonalinclusion,andlet
#: QBHf x QBHf t Q(B(H X H)¢HXH) t QBHf
whereta is the transferassociatedto 1\. Then
THEOREM 3.11 (SZHULTZ [11]). The composition
pairingo is homotopic
(undertheidentification
) to the composite
4. The homology of the U(1)-transfer. Let U(1) act by multiplication
on
S2m-1c cm. Thereresults(2.3)a stabletransfermap
cp+m-1AS1 ) S2m-1
usingthecanonical
trivialization
of theadjointbundleto identifythesource.Pinching S2m-1to a pointgivesan "Eulerclass"CP+m-1
A S1 ) S°. Thesearecompatibleoverm, andgivea stablemap
This is preciselythe map consideredby K. Knappin [6]. Wewill computethe
homologyof its adjoint,the pointedmap
Actually,the sameworkyieldsa computationof the homologyof the transfer
associatedto an inclusionZpnc U(1). If LndenotesBZpnx
thenthis is a pointed
map
(4 3)
tn:CP+°°
AS1 , QLn+.
To state the results,let H* denotemodp homology,p any prime. Let x E
H2(CP°°) be the canonicalgenerator,let ar E H2r(CP°°)be dualto xr, andlet
ar E H2r+l(CP+°°
AS1) be the suspension
of ar. Letu E Hl(Ln) andv = -:nu E
H2(Ln)
be thenaturalgenerators,
andlet er E Hr(Ln)be the dualof themonomial
x backto v; then 7r*e2n
= an
THEOREM4.4 . Letp = 2 andlet tn beas in (4.2) and(4.3). Then
(a) to*ar= E ¢Q28+1[1]
* XQ2(r8)[1]+ St Qt+lQt[l] * (xQr t[l])*2.
(b) tl*ar= E 3e28+l* xe2(r_s)+ St Qt+let * (xer-t)*2.
(c) Forn > 2, tn*ar
= E 3e2s+l* xe2(r_s)+ St Q2t+le2t * (Xer-2t)*2
THEOREM4 . 5 . Letp beoddandlet tn beas in (4.2) and(4.3). Then
(a) tO*ar = (-l)k Et dQt[l] * XQk-t[I] for r = (p- 1)k- 1, and= Ootherwise.
(b) Forn > 1, tn*ar= , e28+l* xe2(r_8).
Ln+ A S1 CP+°°
A S
S1-EQUIVARIANT FUNCTION SPACES
245
PROOF. WeapplyLemma2.7 to Zpn
c U(1) with E = S2m-1. As a framed
manifold,the quotientgroupU(l)/Zpn
is S1withits nonbounding
framing.The
lemmathusadjointsto give(takingm = oo henceforth)
a homotopy-commutative
diagram:
lA[Sl]
S
Ln+
AQS+
(4.6)
ol tn
At
Q((LnAS1)+) Q+
QL+
HereA is the usualsmashproductmap,anda is the actionmap.Since7r*e2r
= ar,
we cancomputetn*arfrom(4.6).
To beginwith,let
h:1r*QS+,HQS1
be the Hurewiczhomomorphism.
Thenh[Sl]E H1QS+is characterized
by three
properties:
(1) It is a coalgebra-primitive
andis killedby all Steenrodoperations.
(2) It reducesin H1QS° to the Hurewiczimageof 71E 1rlQS°,by Note 1.11.
Thisis Q1[1]* [-2] forp = 2 and0 forp odd.
(3) It reducesin H1Sl to the fundamental
classa.
It followsthat
(4.7)
h[Sl]= a * [-1] + Q1[l]* [-2] forp = 2,
(4.8)
h[S1]=a*[-l] forpodd.
Wenowrestrictattentionto p = 2, leavingthe ratherdegenerate
caseof p odd
to the reader.Wefirsttreat the casen = 1, so that the diagonaland Steenrod
actionin H*L1= H*RP°°aregivenby
.9)
i\er= E eE)et,
s+t=r
(4.10)
Sqtier= ( t ) er_t.
Recallthe distributivity
formulae[5, p. 15]:
(4.11)
x A (y * z) = ,(x'
A y) * (x" A z),
(4.12)
x A Q y = EQ
((Sq*x)A y),
wherei\x = Ex'X x". From(4.7)and(4.11),
er A h[Sl]= erA (a * [-1] + Q1[l]* [-2])
= ,(es
A a) * (er_ A [-1]) + ,(es
A Ql[l]) * (er_ A [-2]).
t
t
t
B. M. MANN, E. Y. MILLER AND H. R. MILLER
246
From(4.11),(4.12)and (4.10),
er_ A [-1] = Xer-)
er_¢A [-2] = (xet) 2 if r-s-2t,
=0
if r-sisodd,
e,
AQ1[1]= tet*2
if s = 2t-15
= Qt+1et if s-2t.
Thus(replacing
r by 2r)
(4.13)
e2rAh[Sl]= ,(es 2) a) * xe2r_49 + E Qt+let
*
(xer_t)*2.
Finally,in cohomology
a*x = x X 1 + 1 X x,
so dualizing,
(4.14)
*(er X 1) = er,
*(er C)a) = (r + l)er+l.
Theorem4.4(b) now followsby applyinga* to (4.13)and recallingthe diagram
(4.6).
Theproofforp = 2,n > 1, is completelyanalogous,
usingthe differentstructure
of H*Ln,namely
i\e2r
=
E
e28 C) e2t,
t3+t=r
i\e2r+l=
E
e¢@etX
t3+t=2r+
1
Sqt*er
= ( t ) er_t fort even,
=0
fort odd.
Part (a) followsfrompart (b), sincetransferscomposewell (Note 1.15),while
thetransfert: RP+°° ) QS°is givenin homologyby
Q [1]
bydefinitionof the Dyer-Lashof
operationQr Thiscompletesthe proof. O
REMARK4. 15. The oddcellsin BZ+n
mapto 0 in CP+°°,andhencein QBZp+n
andin QS° Thisresultsin the followingamusingidentitiesforall n > 1 andr > 0:
(4.16)In H*(QBZ2+n;
Z2)X
t*er =
E
e243+l * xe2(r-43)+l
+ E
e2t2+l * (Xer-2t)
2= 0
(4.17)In H*(QS°;
Z2)
E
Q2s+1[1]
* XQ2(r-8)+1[1]
+
E(Q2t+
[1]) *(XQ [1])
(4.18)In H*(QBZp+n;
Zp)forp odd,
E
e24¢+l * xe2(r_8)+l
= °-
tary
l:
to
transfer
Q(CP+
(a)
X
1.)transformations
A
Define
QX
jZ2
S1)
tn:
denote
AR:
°QBZ+nx
aCP+
AR
map
RP
the
AXt:
is
CP+n-1
S1
)standard
are
by
QBZ+n
RP+
0)Theorem
S1-equivariant.
AQRP+
AS1
In
inclusion,
RP
U(n)
homology
3X10
b ) by
S0)
the
sending
and
itDefine
infinite
Ac
therefore
i\: CP+
(I,z),
XaXmap
loop
x ASl
X
commutes
AR:
where
the
extension
RPn-l
U.
diagonal
I C
with
O(n)
Cn
ofis
loop
map,
the
bya
S1-EQUIVARIANT FUNCTION SPACES
247
(4.19)In H*(QS°;
Zp)forp odd,
/3Qt[1]
*X/3Qrt[1]= 0
E
t
REMARK4. 20.
A similaranalysismaybe appliedto computethe transfer
QRP+
t : (HP°°)¢,
in mod2 homology,
whereHP°° is quaternionic
projectivespaceand Z2 C Sp(1)
is the center.TheThomcomplex(HP°°)¢ is thenJames'"quasiprojective
space".
The computationis substantially
moretedioushere,however,sinceby the same
argumentas above,the Hurewiczimageof the stablehomotopyfundamental
class
of Sp(1)/Z2= S0(3)in H3Q(S0(3)+)
hasnineterms:
h[S0(3)]= e3
+
el
* e2 * eO 1 +
+Q3eo*eol
+el
+ Q eO * Qlel
* eO
Q2el
* eO 1
*Qlel
3+
*eO2+Q2Qleo
QleO
* Q2eo
* eO
*eO3
3+ Q1eo* QlQ1
-7
whereen generatesHnSO(3).
REMARK 4.21. The restrictionmap G(S1) ) G(Zpn))
regardedas a map
productsandloopDyer-Lashof
operations.SincethesegenerateH*(Q(CP+
AS1))
fromH*(XP+ AS1), we havecompletelyanalyzedthe homologyof the restriction
mapsG(S1) ) G(Zpn
) andG(S1) >G(1)
5. EquivariantJ-homomorphisms. Weshallstudymaps
jz2
°
>
QRP+X
jSl
U
)
Q(CP+AS1)
arisingfromthe fact that orthogonal
transformations
are Z2-equivariant
andunisendinga realline I c R.nto the reflectionthroughthe hyperplane
perpendicular
complexline and z E C has lZl = 1, to the unitarytransformation
whichis the
identityon 1l andmultiplication
by z on 1. Also,let A:RPn-l ) SO(n)be AR
composedwith the reflectionR throughthe hyperplane
x1 = 0 (if x1,. . ., xn are
the coordinates
in Rn). Thesemapsarecompatibleas n varies,andgivemaps
Leti: S° ) RP+ be inducedby the inclusion1 C Z2, andlet t: RP+ ) QS°
be the associatedtransfer.Let t>: RP+ ) QRP+be the transferassociatedto
the identitymapof RP°°with the framingtwistedby A (as in Note 1.17). Any
infiniteloopspacehas an involutionX obtainedby smashingwith -1 E QS° Let
foranyspaceX.
THEOREM5.1.
Withthesenotations,
(b) Thefollowingthreemapsarehomotopic.
The
QRP+°°
PROOF
adjoint
is homotopic
OFmap
5.1(a)
S2n-l/Z2
f: toRP2n-l
itX.
Let
QS°-AR:
By
x s2n-l
Theorem
RP2n-1
QRP+°°
s2n-l
3.11,
S2n-l/Z
0(2n)
istherefore,
smooth,
be AR
and
the
composed
the
lower
associated
right-hand
with the
map
B. M. MANN, E. Y. MILLER AND H. R. MILLER
248
(i)RP+
A SO JZ2 QRP+,
(ii)RP+A&QRP+XQRP+
(iii)RP+
tA
ttX
QRP+
()QRP+
QRP+*(1)QRP
(C)jS1OACt:CP+AS1Q(CP+AS1).
Beforegiving the proofs,we note some corollaries.
COROLLARY
5.2. In the stablecategory,CP+°°AS1 is a retractof U. a
COROLLARY
5.3. (a) In mod2 homology,
jZ2er =
E
Q8eo *xet *eOl.
s+t=r
(b) In homology
withanycoefficients,
jSl*ar
=
ar.
2
Herewe are writinger for A*erand ar for AC*ar
antipodalmap A E S0(2n). Since S0(2n) is connected,->R
Weapply (3.5) to the map
f
RP2n1
JZ2 ° (-aR):
f/ is transverseto the diagonali\
/
P (
> Endz
1
(S2n-1)
1t
X S2n-1/z2
<
homotopicto AR.
(see (3.5)), with pull-backs2n-1/Z2
1^
RP2n-l
iS
g
(s2n-l
x s2n-l)lz2
prl
> Rp2n-1
Thecompositep is clearly the identity. The framing,however,is nontrivial;it is
twistedbecauseIo(A)is pulledback acrossthe degree -t map f. Thus
jZ2°>RXt
byNote 1.17.
The proofof 5.1(c) is similar,except that the analogueof f/ has degree+1.
PROOFOF5.1(b). By Lemma2.8, we have a homotopy-commutative
diagram
QS° x QRP+
],lAt
ix1
QRP+ x QRP+°°
Jr#
where
# is the diagonaltransferas in Theorem3.11. Thus the map (-1)#: QRP+°°
RP+
RP+ QRP+
QRP+
/ U
QRP+
Q(BW+)K)
CP+
x QRP°°
Q(S
AA(H),
it-*X
S t ) QRP°°
S1-EQUIVARIANT FUNCTION SPACES
249
boxin the followingdiagramcommutes.
QRP+ x QRP+
4
>
t
AR
(
0
X
>, Ro
<
o
t
t Xxx
X
QRP
JZ2
QRP+ x QRP+
>, (-l)o
QRP
JZ2
>
t itX*l
*l
) it*x
v
-1)
QRP++
The upperleft-handbox commutesby (S.l)(a), andthe rest of the diagramcommutesfortrivialreasons.Thisproves(i)(ii).
To see that (ii)(iii), we recallthat E. H. Brownhas proven[4] that tx is
homotopicto the composite
Theresultfollows. L
Theorem3.10impliesthat
-
u
/
c(
S
JS1
Q(CP+
A S1)
Jc
Q1(S°)
k
t*(-1)
iQo(S°)
commutesup to homotopy.Adjointto this is the stable diagram:
(5.4)
S jc
so
S°
id
)
TheoremC of the Introduction
followsfromthis and Theorems4.4 and 4.5.
Also,(5.4)togetherwithS.l(c) yieldthe homotopycommutative
stablediagram:
CP+ AS1
S
S°
t
U
>
jc
S°
t
id
6. The local structure of H*(SS(Sl);Zp). In this sectionwe describeformulaesufficientto characterize
the R-Hopfalgebrastructureof H*(SS(S1);Zp).
Werelyon [8]. Therewe construct,fora finitegroupH, an EOO-ring
spaceA(H),
calledthe Burnsidespaceof H. Thereis a homotopyequivalence
of infiniteloop
spaces
(6.1)
K
whereK rangesovera set of subgroups
of H representing
its conjugacyclassesof
subgroups,
andWkis the quotientby K of the normalizer
of K in H. Thereis thus
a map
(6.2)
i: QBH+ , A(H),
250
B. M. MANN, E. Y. MILLER AND H. R. MILLER
whichis up to homotopya split monomorphism.
The ring-structure
map ,u in
A(H)extendsthe transfer# inducedas in Theorem3.11by the diagonalinclusion
A: H ) H x H; that is to say,the diagram
(6.3)
QBH+x QBH+ #
L ixi
A(H)x A(H) 4
QBH+
,i
A(H)
is homotopy-commutative.
If H admitsa finite-dimensional
orthogonal
representation
W suchthat the unit
spheresW is a freeH-space,then a mapp: G(H) ) A(H) is constructed.This
mapcarriesthe infiniteloop-spacestructure[3, 8] on SG(H)to the multiplicative
structurein A(H). Furthermore,
if *(-1) denotesthe evidentcomponent-shifting
self-mapof A(H), then
G(H) ;
QBH+
PS
S i
A(H) *(-1) A(H)
is homotopy-commutative.
In particular,p is (upto homotopy)a split monomorphism.
It followsthatrelationsbetweenthecomposition
structureandtheloopstructure
in H*(SG(H))
can be obtainedby translatingthe usualdistributivity
andmixed
CartanandAdemrelationsvalidin H*(A(H)).To describethe translation,let
# F*(A(H))C)H*(A(H)) ) H*(A(H))
bethe multiplicative
product,u*;on elementsin the imageof H*(QBH+)
it is inducedby the diagonaltransfer.LetQrbe the multiplicative
Dyer-Lashof
operation
inH*(A(H)),
while* andQrdenotethe additive(loop)productandDyer-Lashof
operation.
Leto andQrdenotethe (composition)
productand(composition)
DyerLashofoperationin SG(H).Then
LEMMA
6. 5 . Omitting
"p*
",
x ° Y= (-1)1S
Qr(x* y) =
E
1IYIx'* y' * (x//#y//)
Qix'* Qj(x #y ) * Q (y ),
i+j+k=r
where
/\x = EX'@ x" and/\y = Sy'@ y".
PROOF. The first expressionis due (in the moregeneralcase of a periodic
compact
Liegroup)to R. E. Schultz[11];it followsfromhis Theorem3.11above.
Itfollowsalsofrom(6.3)andHopf-ring
distributivity:
(x o y) * [1]= (x * [l])#(y * [1])
= (-1)1x 1IYIx'* y' * (x"#y")* [1].
Thesecondexpression
followssimilarlyfrom(6.3),the mixedCartanformula,and
thefactthat QU[1]= Oforu > O. O
X H*(CP°°)
THEOREM
6.6.
H*(CP°°)
Let is
p =the
2. identity
In H*(SG(S1))
(with mod2
H*(Q(CP+
coefficients),
/\ S1)), and that
251
S1-EQUIVARIANT FUNCTION SPACES
space
In principle,the structuralformulaevalidin the homologyof an EOO-ring
couldnowbe translatedintoformulaerelatingo andQrto * andQr.Thesewould
determinethe structureof H*(SG(H)),givex o y andQrxforx, y C H*(BH). In
viewof the complexityof the mixedAdemrelation,however,we shallnot complete
this exercisehere.
Weturnnowto S1.
Sincethe infiniteloopmapi: SG(S1) > SG(Zp)is injectivein homology,the
aboveholdalsoin H*(SG(S1 )) . Indeed,sincei* wascomputed
considered
formulae
in §4,whilethe R-Hopfalgebrastructureof H*(SG(Z2)) wascomputedin [8],we
the R-Hopfalgebrastructureof H*(SG(S1); Z2); andsimilarly,forp odd,
"know"
the Hopfalgebrastructureof H*(SG(Sl);Zp) and someinformation
we "know"
aboutits R-modulestructure.Wewishto be moreexplicit,however;and,by the
whereas in §4, ar
indicatedformulae,it will sufficeto computeaq o ar andQqarX
is the canonicalgeneratorof H2r+l(CP+ /\ S1;Zp).
(a)
- s, rS)Q2(q+r-S)+la
aq o ar = aq * ar + ,(q
1Qq+r-2ta
(q-2t, r-2t) Qq+r+
+,
t
Q
(b)
THEOREM6.7.
(a)
r
r )
(
q+rn
Q2q+la
Letpbeodd. InH*(SG(Sl))_
=°
H*(Q(CP+ASl)),
aq o ar = aq * ar + E c(q,r, t):Qtaq+r+l_(p- l)t t
where
(q' ' )
(b)
(
r
(
)
Qqar= (-l)q+r+l (q r
PROOF OF THEOREM6.6.
) (
t- k
) ar+q(p-l)
We applyLemma2.9 with H = S1. Note that
/\ S1) > H*(CP+/\ S1) is givenby
*: H*(CP+) (23H*(CP+
,i* (aU X ar)-(U,
r)ar+u
Thus,
(6.8)
)
aq#ar = E (u, r)aq-u/\ t*ar+U
u
B. M. MANN, E. Y. MILLERAND H. R. MILLER
252
to Theorem4.4, t*ar+Uis a sumof twosortsof terms.Forthe first,we
According
use
= aq-u X 1 + 1 8) a
i\aq-u
1 /\ XQt[l] = bo,
( q- u - t ) _
2t-
to find
(6.9)
[1])
[1]* XQ2(U+r-8)
aq-u/\ (Q28+1
= bu+r
E
(
q
u t)
Q2s+2t+laq-u-t
forthe secondwe find
Similarly,
aq-u
* (XQU+r-8[1])*2)
/\ (Q8+lQ8[l]
E
(
t
) (
t
) Qs+2i+1Qs+2ta
NowQ8+2i+1killsthis dimensionunless2i > q - u, whilethe firstbinomialcoeffiis thustrivialunlessq - u is even,
cientis zerounless2i < q - u. The expression
andthen
* (XQU+r-8[1])*2)
a2V/\ (Q8+lQ8[1]
(6.10)
= bu+r
E
t)
( V
Q8+q-u+l
Q82tav-t
a
Substituting(6.9)and(6.10)into (6.8),
aq#ar= (u,r)
(q
+ (q-2v,
t
t)
t
r) (
Q2(u+r+t)+la
) Qq+r+lQq+r-2(t+t)av-t
v,t
Now,forfixeds,
E
u+t=q-s
E
v-t=s
t
(u,r) (
(q-2v,r) (
) = (q- s)r - s),
t ) = (q-2s,r-2s),
so we concludethat
aq#ar= ,(q-
s,r - s)Q2(q+r-8)+la
+ ,(q-2s,
r-2s)Qq+r+lqq+r-2sa
Part(b) follows fromKochman's
w: H*(QBZp+;
formula
toF
[7]
H*
Zp)
and
(QS°
Nthe fact
) that jsl: U SG(S1)
S1-EQUIVARIANT FUNCTION SPACES
253
Nowaqandar areprimitive,so by Lemma6.5,
aq o ar = aq * ar + aq#ar,
andpart(a) of the theoremfollows
is an infinite-loop
map[3]. 0
Theproofof Theorem6.7 is analogousandis left to the reader. O
REMARK
6.11. From(6.6)we findthat
ar
°
ar
=
ar
*
ar
+
Q2r+lar
=
°
whichis consistentwithCorollary
5.3. Indeed,it canbe usedto provethatis1* ar =
ar at p= 2.
7. The global structure of H*(SG(S1))and of H*(BSG(S1)).Recallthe
weightval?lation
w:H(QS°;z)N
whereN = {0,1, . . ., x}. It is the smallestfunctionwith this sourceandtarget
satisfying
(7.1)
w(x * y) > w(x)+ w(y),
(7.2)
w(x + y) > min{w(x),w(y)},
W(QI [1]) = pl(I)
forI admissible
withe(I) > O.
(Wereferto [5, pp. 16, 42]fordefinitionsoflength,admissibility,
andexcess,for
Dyer-LashoSoperations.)
Thusin particular
w(O)= x andw([n])= O.
Similarly,
we define
respectively
w: H*(Q(CP+°°/\ S 1); Zp) >N,
to be the smallestsuchfunctionssatisfying(7.1),(7.2),and,forI admissible,
w(QIer)
= pl(I)+l
fore(I) > r > O, andl(I) > O,
w(er)
= p
forr > O,
W(eO) = °,
respectively,
W(QIar)
= pl(I)+l
fore(I) > 2r + 1 > 1.
In all threecasesit is clearthat
(7 3)
w(x1* * xr) = w(xl) + + w(xr)
if x1 * * xr is nonzeroandeachxi is suchan element,QI[1],QIer,or QIar.
It is easyto check,usingTheorems4.4 and4.5, that the transfermap
H*(Q(CP+/\S ))
wH*(QBZp+)
],tF
preserveweight:w(0*(x))> w(x) for 0 = to,t1, or t. Thusit makessenseto
computethemmodulohigherweight,andwe havethe followinglemma.
t1*: H*(Q(CP+
A sl)) H*(QB:Z+)
B. M. MANN, E. Y. MILLER AND H. R. MILLER
254
LEMMA7.4.
Foranyp,
tl*QIaT--QIe2r+l
*e
andforp = 2,
to*Q
ar
_
QIQ2r+l[l]
*
[_21(I)+1]
mod higherweight.
PROOF. ForI empty,these are immediatefromTheorems4.4 and 4.5. The
AdemrelationforQnQO
impliesthatforanyn > Oandanyk E Z, w(Qn[pk])
> p2.
It followsby inductionthat
Q (Q
[1]* [_p]) _
QI Q2r+ l [l] * [_ pl (I )+ l ]
modhigherweight.Theassertionforto*follows,andthe othercaseis similar. O
As a corollary,
we havethe injectivityneededin §6:
COROLLARY7.5. Forp = 2,
to*:H*(Q(CP+°°
/\ S1))
ts injective,andfor p arbitrary,
>
H*(QS°)
is injective. O
Nextwe notethe behaviorof the weightvaluationwithrespectto the composition productin SG(S1) Q(CP+/\ 51).
LEMMA7.6. In H*(Q(CP+
/\ 51)),
(a) w(x#y) > w(x) + w(y), withequalityif andonlyif p = 2 andw(x) = 2 =
w(y); i.e., x = as,y = at
(b) w(xo y) > w(x) + w(y).
(c) x o y--x
* y modulo
higherweightunlessp = 2,x = as,y = at
PROOF. (a) By the mixedCartanformula,QIa¢#QJat
is a sum of termsof
the formQK(ai#a;),
wherel(K) = I(I) + I(J). Substituting
in the valueof a#aj
fromTheorems6.6 and6.7 we findthat eachtermin this sumhassveightat least
pl(I)+l(J)+2.
NOW
pl(I)+l(J)+2
>
pl(I)+1
+ pl(J)+2
andthe inequalityis strictunlessp = 2 and l(I) = O= I(J). Thisproves(a) for
suchelements.If xl, . . ., xq,Y1,. . ., Yr)aresuchelements,theneachis primitive,so
the distributivity
formulaimpliesthat
(xl * * xq)#(y1* * yr) = O if q 7& r
= 2, i(Xl#Ya(l)) * * * (xq#ya(q)) if q = r
aCSq
andthe resultholdsby (7.3).
(b) and (c). SinceQIaS andQJatareprimitive,
QIa,
o QJat
= QIa,
* QJat + QIa,#QJat
by Lemma6.5. By (a), this is congruent
to QIaS * QJat modhigherweight,with
the notedexceptions.Again,the extensionto monomials
is easy. O
H*(SG(S1))
H*(SG(S1))
H*(U) X P[ar*2:
H*(Q(CP+
r >/\ O]sl)),
S1-EQUIVARIANT FUN5TION SPACES
255
we have
As a corollary,
THEOREM7.7. H*(SG(S1)) is a primitivelygeneratedHopf algebra.For p =
2,
X
H*(Q(CP+ /\ s1))//P[ar: r > O],
andfor p odd,
as Hopf algebras. C]
spacespectralsequence
WenowstudyH*(BSG(S1 )) bymeansof the classifying
(7.8)
TorHF (SG(S1)) (Z
Zp) > H*(BSG(S1 ))
first,
homologyspectralsequenceof Hopfalgebras.Consider
Thisis a firstquadrant
with s > 1, the dividedpowersof the suspensionvar of ar.
amongthe generators
is1: U >SG(S1) mapsontotheseelements,by
J-homomorphism
Theequivariant
Corollary
5.3. Sincethe spectralsequence
Tor ( )(Zp,Zp)> H*(BU)
cycles.
arepermanent
collapsesat E2, we concludethat thesegenerators
with s > 1, so the spectralsequence
Forp =-2, thereareno furthergenerators
(7.8) collapsesat E2. Forp 7&2, eachodd generatorin H*(SG(S1)) leadsto a
[5,
dividedpowersequencein E2. Theseareconnectedby the universaldifferential
p. 125]
dP-1ap+j (ax) = - (ff:Q 1x)Aj(5x) .
Here,if 2s =
have
Ixl +
1, thenQ1x = uQSxforsomeunitu E Zp. To computeQ1, we
7.9. In H*(SG(S1)), for I admissiblewithe(I)
LEMMA
>
2r+ 1 and l(I)
>
O,
QlQIar-QlQIar
modhigherweight.
This is a routineexercisewith the weightvaluation,and is describedin more
detailin [9],so its proofis omittedhere.Thislemmaresultsin
EPr[var:
X
r>O]@E[ff:Qsar: str>O]
D[aQIar: l(I)
>
1, e(I)
>
2r + 1, r > O],
algebratruncatedat heightp. No further
whereD denotesthe freecommutative
arenowpossible:EP = E°°.
differentials
andthenextlemma
J-homomorphism,
Lemma7.9, togetherwiththeequivariant
extension,andTheoremF of the Introthe multiplicative
whenp = 2, determines
ductionresults.
B. M. MANN, E. Y. MILLER AND H. R. MILLER
256
LEMMA 7.10.
Forp= 2
Q4r+3Q2r+l
a =O
in H*(SG(S1)) .
PROOF. UsingLemma6.5 andthe factthat ar is primitive,
we expand
Q4r+3
Q2r+ 1 a
= Q4r+3 (a*2)
intoa sumin whicheachtermappearsexactlytwice.Sincewe areworkingmod2,
the lemmafollows. O
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238-261.
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DEPARTMENTOF MATHEMATICS,
CLARKSONUNIVERSITY,POTSDAM,NEW YORK
13676 (Currentaddressof B. M. Mann)
DEPARTMENT
OF MATHEMATICS,
POLYTECHNIC
INSTITUTEOF NEW YORK, BROOKLYN,NEW YORK11201 (Currentaddress of E. V. Miller)
DEPARTMENT
OF MATHEMATICS,
UNIVERSITYOF WASHINGTON,
SEATTLE,WASHING
TON98195
Currentaddress(H. R. Miller): Universitede Paris, UER de Mathematique,2, Place Jussieu,
75251 Paris Cedex 05, France